Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,27,35)(16,28,30,21)(17,22,18,23)(25,39,42,33)(26,34,31,37)(29,38,32,36)(40,44,41,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(16,29)(17,18)(21,36)(22,23)(25,40)(26,31)(28,38)(30,32)(33,43)(34,37)(39,44)(41,42);(2,4)(13,15,19,27)(14,20,24,35)(16,25,30,42)(17,31,18,26)(21,33,28,39)(22,37,23,34)(29,40,32,41)(36,43,38,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,33)(22,36)(23,38)(24,28)(26,41)(27,42)(31,40)(34,44)(35,39)(37,43);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,37)(21,36,28,38)(25,41,42,40)(33,44,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429680);